Theorem mulasssrg 7566

Description: Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)

Assertion

Ref	Expression
mulasssrg	|- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) )

Proof of Theorem mulasssrg

Dummy variables f g h r s t u v w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 7535	. 2 |- R. = ( ( P. X. P. ) /. ~R )
2		mulsrpr 7554	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. z , w >. ] ~R ) = [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R )
3		mulsrpr 7554	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. z , w >. ] ~R .R [ <. v , u >. ] ~R ) = [ <. ( ( z .P. v ) +P. ( w .P. u ) ) , ( ( z .P. u ) +P. ( w .P. v ) ) >. ] ~R )
4		mulsrpr 7554	. 2 |- ( ( ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( [ <. ( ( x .P. z ) +P. ( y .P. w ) ) , ( ( x .P. w ) +P. ( y .P. z ) ) >. ] ~R .R [ <. v , u >. ] ~R ) = [ <. ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. v ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. u ) ) , ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. u ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. v ) ) >. ] ~R )
5		mulsrpr 7554	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( ( ( z .P. v ) +P. ( w .P. u ) ) e. P. /\ ( ( z .P. u ) +P. ( w .P. v ) ) e. P. ) ) -> ( [ <. x , y >. ] ~R .R [ <. ( ( z .P. v ) +P. ( w .P. u ) ) , ( ( z .P. u ) +P. ( w .P. v ) ) >. ] ~R ) = [ <. ( ( x .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) +P. ( y .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) , ( ( x .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) +P. ( y .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) ) >. ] ~R )
6		mulclpr 7380	. . . . 5 |- ( ( x e. P. /\ z e. P. ) -> ( x .P. z ) e. P. )
7	6	ad2ant2r 500	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. z ) e. P. )
8		mulclpr 7380	. . . . 5 |- ( ( y e. P. /\ w e. P. ) -> ( y .P. w ) e. P. )
9	8	ad2ant2l 499	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. w ) e. P. )
10		addclpr 7345	. . . 4 |- ( ( ( x .P. z ) e. P. /\ ( y .P. w ) e. P. ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
11	7, 9, 10	syl2anc 408	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. z ) +P. ( y .P. w ) ) e. P. )
12		mulclpr 7380	. . . . 5 |- ( ( x e. P. /\ w e. P. ) -> ( x .P. w ) e. P. )
13	12	ad2ant2rl 502	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x .P. w ) e. P. )
14		mulclpr 7380	. . . . 5 |- ( ( y e. P. /\ z e. P. ) -> ( y .P. z ) e. P. )
15	14	ad2ant2lr 501	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y .P. z ) e. P. )
16		addclpr 7345	. . . 4 |- ( ( ( x .P. w ) e. P. /\ ( y .P. z ) e. P. ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
17	13, 15, 16	syl2anc 408	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( x .P. w ) +P. ( y .P. z ) ) e. P. )
18	11, 17	jca 304	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( x .P. z ) +P. ( y .P. w ) ) e. P. /\ ( ( x .P. w ) +P. ( y .P. z ) ) e. P. ) )
19		mulclpr 7380	. . . . 5 |- ( ( z e. P. /\ v e. P. ) -> ( z .P. v ) e. P. )
20	19	ad2ant2r 500	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. v ) e. P. )
21		mulclpr 7380	. . . . 5 |- ( ( w e. P. /\ u e. P. ) -> ( w .P. u ) e. P. )
22	21	ad2ant2l 499	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. u ) e. P. )
23		addclpr 7345	. . . 4 |- ( ( ( z .P. v ) e. P. /\ ( w .P. u ) e. P. ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
24	20, 22, 23	syl2anc 408	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. v ) +P. ( w .P. u ) ) e. P. )
25		mulclpr 7380	. . . . 5 |- ( ( z e. P. /\ u e. P. ) -> ( z .P. u ) e. P. )
26	25	ad2ant2rl 502	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( z .P. u ) e. P. )
27		mulclpr 7380	. . . . 5 |- ( ( w e. P. /\ v e. P. ) -> ( w .P. v ) e. P. )
28	27	ad2ant2lr 501	. . . 4 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( w .P. v ) e. P. )
29		addclpr 7345	. . . 4 |- ( ( ( z .P. u ) e. P. /\ ( w .P. v ) e. P. ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
30	26, 28, 29	syl2anc 408	. . 3 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( z .P. u ) +P. ( w .P. v ) ) e. P. )
31	24, 30	jca 304	. 2 |- ( ( ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( z .P. v ) +P. ( w .P. u ) ) e. P. /\ ( ( z .P. u ) +P. ( w .P. v ) ) e. P. ) )
32		mulcomprg 7388	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) = ( g .P. f ) )
33	32	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f .P. g ) = ( g .P. f ) )
34		distrprg 7396	. . . . . 6 |- ( ( r e. P. /\ s e. P. /\ t e. P. ) -> ( r .P. ( s +P. t ) ) = ( ( r .P. s ) +P. ( r .P. t ) ) )
35	34	adantl 275	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. /\ t e. P. ) ) -> ( r .P. ( s +P. t ) ) = ( ( r .P. s ) +P. ( r .P. t ) ) )
36		simp1 981	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> f e. P. )
37		simp2 982	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> g e. P. )
38		simp3 983	. . . . 5 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> h e. P. )
39		addclpr 7345	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r +P. s ) e. P. )
40	39	adantl 275	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r +P. s ) e. P. )
41		mulcomprg 7388	. . . . . 6 |- ( ( r e. P. /\ s e. P. ) -> ( r .P. s ) = ( s .P. r ) )
42	41	adantl 275	. . . . 5 |- ( ( ( f e. P. /\ g e. P. /\ h e. P. ) /\ ( r e. P. /\ s e. P. ) ) -> ( r .P. s ) = ( s .P. r ) )
43	35, 36, 37, 38, 40, 42	caovdir2d 5947	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) .P. h ) = ( ( f .P. h ) +P. ( g .P. h ) ) )
44	43	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f +P. g ) .P. h ) = ( ( f .P. h ) +P. ( g .P. h ) ) )
45		mulassprg 7389	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f .P. g ) .P. h ) = ( f .P. ( g .P. h ) ) )
46	45	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f .P. g ) .P. h ) = ( f .P. ( g .P. h ) ) )
47		mulclpr 7380	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f .P. g ) e. P. )
48	47	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f .P. g ) e. P. )
49		simp1l 1005	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> x e. P. )
50		simp1r 1006	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> y e. P. )
51		simp2l 1007	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> z e. P. )
52		simp2r 1008	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> w e. P. )
53		simp3l 1009	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> v e. P. )
54		simp3r 1010	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> u e. P. )
55		addcomprg 7386	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
56	55	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
57		addassprg 7387	. . . 4 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
58	57	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( ( f +P. g ) +P. h ) = ( f +P. ( g +P. h ) ) )
59		addclpr 7345	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) e. P. )
60	59	adantl 275	. . 3 |- ( ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) e. P. )
61	33, 44, 46, 48, 49, 50, 51, 52, 53, 54, 56, 58, 60	caovlem2d 5963	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. v ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. u ) ) = ( ( x .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) +P. ( y .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) ) )
62	33, 44, 46, 48, 49, 50, 51, 52, 54, 53, 56, 58, 60	caovlem2d 5963	. 2 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) /\ ( v e. P. /\ u e. P. ) ) -> ( ( ( ( x .P. z ) +P. ( y .P. w ) ) .P. u ) +P. ( ( ( x .P. w ) +P. ( y .P. z ) ) .P. v ) ) = ( ( x .P. ( ( z .P. u ) +P. ( w .P. v ) ) ) +P. ( y .P. ( ( z .P. v ) +P. ( w .P. u ) ) ) ) )
63	1, 2, 3, 4, 5, 18, 31, 61, 62	ecoviass 6539	1 |- ( ( A e. R. /\ B e. R. /\ C e. R. ) -> ( ( A .R B ) .R C ) = ( A .R ( B .R C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480  (class class class)co 5774   P.cnp 7099   +P. cpp 7101   .P. cmp 7102   ~R cer 7104   R.cnr 7105   .R cmr 7110

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-iplp 7276  df-imp 7277  df-enr 7534  df-nr 7535  df-mr 7537

This theorem is referenced by:  axmulass  7681